scholarly journals STOCHASTIC STABILITY OF NON-UNIFORMLY HYPERBOLIC DIFFEOMORPHISMS

2007 ◽  
Vol 07 (03) ◽  
pp. 299-333 ◽  
Author(s):  
JOSÉ F. ALVES ◽  
VÍTOR ARAÚJO ◽  
CARLOS H. VÁSQUEZ
Keyword(s):  
Stochastic Stability ◽  
Sufficient Conditions ◽  
Random Perturbations ◽  
Dominated Splitting ◽  
Hyperbolic Diffeomorphisms ◽  
Stable Direction ◽  
Stochastically Stable ◽  
Necessary And Sufficient ◽  
Open Class ◽  
Unstable Direction

We prove that the statistical properties of random perturbations of a diffeomorphism with dominated splitting having mostly contracting center-stable direction and non-uniformly expanding center-unstable direction are described by a finite number of stationary measures. We also give necessary and sufficient conditions for the stochastic stability of such dynamical systems. We show that a certain C2-open class of non-uniformly hyperbolic diffeomorphisms introduced by Alves, Bonatti and Viana in [2] are stochastically stable.

scholarly journals Strong stochastic stability for non-uniformly expanding maps

2012 ◽  
Vol 33 (3) ◽  
pp. 647-692 ◽  
Author(s):  
JOSÉ F. ALVES ◽  
HELDER VILARINHO
Keyword(s):  
Stochastic Stability ◽  
Hyperbolic Systems ◽  
Sufficient Conditions ◽  
Unperturbed System ◽  
Random Perturbations ◽  
Expanding Maps ◽  
Srb Measure ◽  
Central Direction ◽  
Open Class ◽  
Discrete Time Dynamical Systems

AbstractWe consider random perturbations of discrete-time dynamical systems. We give sufficient conditions for the stochastic stability of certain classes of maps, in a strong sense. This improves the main result in Alves and Araújo [Random perturbations of non-uniformly expanding maps. Astérisque 286 (2003), 25–62], where the stochastic stability in the $\mathrm {weak}^*$ topology was proved. Here, under slightly weaker assumptions on the random perturbations, we obtain a stronger version of stochastic stability: convergence of the density of the stationary measure to the density of the Sinai–Ruelle–Bowen (SRB) measure of the unperturbed system in the $L^1$-norm. As an application of our results, we obtain strong stochastic stability for two classes of non-uniformly expanding maps. The first one is an open class of local diffeomorphisms introduced in Alves, Bonatti and Viana [SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140 (2000), 351–398] and the second one is the class of Viana maps.

scholarly journals ON INVERSE STOCHASTIC RECONSTRUCTION PROBLEM

2021 ◽  
Vol 335 (1) ◽  
pp. 6-13
Author(s):  
M.I. Tleubergenov ◽  
G.K. Vassilina ◽  
A.T. Sarypbek
Keyword(s):  
Integral Manifold ◽  
Sufficient Conditions ◽  
Linear Case ◽  
Inversion Method ◽  
Random Perturbations ◽  
Reconstruction Problem ◽  
Stochastic Problem ◽  
Independent Increments ◽  
Processes With Independent Increments ◽  
Necessary And Sufficient

In this paper, general reconstruction problem in the class of second-order stochastic differential equations of the Ito type is considered for given properties of motion, when the control is included in the drift coefficient. And the form of control parameters is determined by the quasi-inversion method, which provides necessary and sufficient conditions for existence of a given integral manifold. Further, the solution of the Meshchersky’s stochastic problem is given, which is one of the inverse problems of dynamics and, according to the well-known Galiullin’s classification, refers to the restoration problem. It is assumed that random perturbations belong to the class of processes with independent increments. To solve the posed problem an equation of perturbed motion is drawn up by the Ito rule of stochastic differentiation. And, further, the Erugin method in combination with the quasi-inversion method is used to construct: 1) a set of control vector functions and 2) a set of diffusion matrices that provide necessary and sufficient conditions for a given second-order differential equation of Ito type to have a given integral manifold. The linear case of a stochastic problem with drift control is considered separately. In the linear setting, in contrast to the nonlinear formulation, the conditions of solvability in the presence of random perturbations from the class of processes with independent increments coincide with the conditions of solvability in a similar linear case in the presence of random perturbations from the class of independent Wiener processes. Also considered is the scalar case of the recovery problem with drift controls.

scholarly journals ON INVERSE STOCHASTIC RECONSTRUCTION PROBLEM

2021 ◽  
Vol 335 (1) ◽  
pp. 6-13
Author(s):  
M.I. Tleubergenov ◽  
G.K. Vassilina ◽  
A.T. Sarypbek
Keyword(s):  
Integral Manifold ◽  
Sufficient Conditions ◽  
Linear Case ◽  
Inversion Method ◽  
Random Perturbations ◽  
Reconstruction Problem ◽  
Stochastic Problem ◽  
Independent Increments ◽  
Processes With Independent Increments ◽  
Necessary And Sufficient

In this paper, general reconstruction problem in the class of second-order stochastic differential equations of the Ito type is considered for given properties of motion, when the control is included in the drift coefficient. And the form of control parameters is determined by the quasi-inversion method, which provides necessary and sufficient conditions for existence of a given integral manifold. Further, the solution of the Meshchersky’s stochastic problem is given, which is one of the inverse problems of dynamics and, according to the well-known Galiullin’s classification, refers to the restoration problem. It is assumed that random perturbations belong to the class of processes with independent increments. To solve the posed problem an equation of perturbed motion is drawn up by the Ito rule of stochastic differentiation. And, further, the Erugin method in combination with the quasi-inversion method is used to construct: 1) a set of control vector functions and 2) a set of diffusion matrices that provide necessary and sufficient conditions for a given second-order differential equation of Ito type to have a given integral manifold. The linear case of a stochastic problem with drift control is considered separately. In the linear setting, in contrast to the nonlinear formulation, the conditions of solvability in the presence of random perturbations from the class of processes with independent increments coincide with the conditions of solvability in a similar linear case in the presence of random perturbations from the class of independent Wiener processes. Also considered is the scalar case of the recovery problem with drift controls.

An Approach, Via Entropy, to the Stability of Random Large-Scale Sampled-Data Systems Under Structural Perturbations

1982 ◽  
Vol 104 (1) ◽  
pp. 49-57 ◽  
Author(s):  
Guy Jumarie
Keyword(s):  
Stochastic Stability ◽  
Large Scale ◽  
Sufficient Conditions ◽  
Direct Application ◽  
Jacobian Determinant ◽  
Random Perturbations ◽  
Data Systems ◽  
Sampled Data ◽  
Large Scale Systems ◽  
The Stability

The concept of entropy in information theory is used to investigate the sensitivity and the stability of sampled-data systems in the presence of random perturbations. After a brief background on the definition, the practical meaning and the main properties of the entropy, its relations with asymptotic insensitiveness are exhibited and then some new results on the sensitivity and the stochastic stability of linear and nonlinear multivariable sampled data systems are derived. A new concept of stochastic conditional asymptotic stability is obtained which seems to be of direct application in the analysis of large-scale systems. Sufficient conditions for stability are stated. This approach provides a new look over stochastic stability. In addition, variable transformations act additively on entropy, via Jacobian determinant, and as a result the corresponding calculus is very simple.

scholarly journals Dissipative Analysis and Synthesis of Control for TS Fuzzy Markovian Jump Neutral Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
R. Sakthivel ◽  
M. Rathika ◽  
Srimanta Santra
Keyword(s):  
Stochastic Stability ◽  
Sufficient Conditions ◽  
Existence Condition ◽  
Linear Matrix ◽  
Control Effort ◽  
Special Cases ◽  
Analysis And Synthesis ◽  
Stochastically Stable ◽  
Dissipative Control ◽  
Takagi Sugeno

This paper is focused on stochastic stability and strictly dissipative control design for a class of Takagi-Sugeno (TS) fuzzy neutral time delayed control systems with Markovian jumps. The main aim of this paper is to design a strictly dissipative controller such that the closed-loop TS fuzzy control system is stochastically stable, and also the disturbance rejection attenuation is obtained to a given level by means of theH∞performance index. Intensive analysis is carried out to obtain sufficient conditions for the existence of desired dissipative controller which ensures both the stochastic stability and the strictly dissipative performance. The main advantage of the proposed technique is that it is possible to obtain the dissipative controller with less control effort and also, as special cases, robustH∞control with the prescribedH∞performance under given constraints and passivity control can be obtained for the considered systems. Also, the existence condition of the fuzzy dissipative controller can be obtained in terms of linear matrix inequalities. Finally, a practical example based on truck-trailer model is provided to demonstrate the effectiveness and feasibility of the proposed design technique.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Sign in / Sign up

Export Citation Format

Share Document